To appear at the International Conf. on Biometrics (ICB), June 2009
Beyond Minutiae: A Fingerprint Individuality
Model with Pattern, Ridge and Pore Features
Yi Chen1,2 and Anil. K. Jain1
1
Michigan State University
2
DigitalPersona Inc.
Abstract. Fingerprints are considered to be unique because they contain various distinctive features, including minutiae, ridges, pores, etc.
Some attempts have been made to model the minutiae in order to get
a quantitative measure for uniqueness or individuality of fingerprints.
However, these models do not fully exploit information contained in nonminutiae features that is utilized for matching fingerprints in practice.
We propose an individuality model that incorporates all three levels of
fingerprint features: pattern or class type (Level 1), minutiae and ridges
(Level 2), and pores (Level 3). Correlations among these features and
their distributions are also taken into account in our model. Experimental results show that the theoretical estimates of fingerprint individuality
using our model consistently follow the empirical values based on the
public domain NIST-4 database.
Key words: fingerprint individuality, statistical models, probability of
random correspondence (PRC), minutiae, ridges, pores
1
Introduction
A number of court challenges have brought into question the validity and reliability of using fingerprints for personal identification [1–3]. These challenges
are based on, among other factors, the lack of (i) conclusive evidence to support the claim of fingerprint uniqueness, and (ii) scientific evaluation of criteria
used to determine a match between two fingerprints. These legal challenges have
generated substantial interest in studies on fingerprint individuality.
Fingerprint individuality can be formulated as the probability that any two
prints from different fingers will be “sufficiently” similar. Because similarity between fingerprints is often quantitatively defined based on the similarity of fingerprint features, it is equivalent to finding the probability of random correspondence (PRC), or the probability of matching k features, given that two impostor
fingerprints contain m and n features, respectively. The key to calculating PRC
is to model the distribution of fingerprint features, as it is not only the number
of features, but also their spatial distributions that account for individuality.
The variety of features that are incorporated in the individuality model is also
important. Latent experts use three levels of features, namely, Level 1 (e.g., pattern), Level 2 (e.g., minutiae and ridges) and Level 3 (e.g., pores) in fingerprint
matching [4]. All of these features are believed to be distinctive and unique and
should be considered in the individuality model (see Fig. 1).
To appear at the International Conf. on Biometrics (ICB), June 2009
2
Yi Chen and Anil. K. Jain
ridge
pore
minutiae
rige loop type whorl type
(a) (b)
Fig. 1. Fingerprint features (a) two different pattern types (b) minutiae, ridges and
pores.
To learn the spatial distribution of fingerprint features, a couple of generative
models have been proposed [5, 6]. Pankanti et al. [5] modeled the minutiae as
uniformly and independently distributed, however, it has been empirically determined that minutiae tend to cluster where ridge directions change abruptly, with
higher density in the core and delta regions [7, 8]. To account for this clustering
tendency of minutiae, Zhu et al. [6] proposed a finger-specific mixture model,
in which minutiae are first clustered and then independently modeled in each
cluster. In modeling features other than minutiae, Fang et al. [9] modeled ridges
by classifying the ridge segment associated with each minutia into one of sixteen
ridge shapes. However, most of the ridge shapes used in this model were quite
rare as the ridge segments associated with minutiae either terminate or bifurcate. Roddy and Stosz [10] proposed to approximate the spatial distribution of
pores using a discrete grid, and consequently, their formulation is sensitive to
distortion as well as the size and the position of the grid cells.
2
Proposed Model
We propose to evaluate fingerprint individuality by modeling the distribution
of minutiae, ridge and pore features. In particular, the minutiae and ridge features are separately modeled for each fingerprint pattern type (whorl, left loop,
right loop, arch and tented arch). This is because minutiae and ridge feature
distributions are highly correlated with the fingerprint pattern [8], which was
also demonstrated by Zhu et al [6]. Compared to finger-specific models, patternspecific models can be easily generalized to different target populations and are
less prone to overfitting, especially when the number of minutiae in a fingerprint is small. To model ridges and pores, transform-invariant features such as
ridge period, ridge curvature, and pore spacing associated with each minutia are
derived.
2.1 Pattern-specific Minutiae Modeling
Given a large database of fingerprints, we assign each image to one of the five major pattern types [11]: whorl, left loop, right loop, arch and tented arch. For each
fingerprint class, minutiae from fingerprints in that class are consolidated and
their spatial distribution is estimated. These minutiae are grouped into clusters
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length
3
based on their positions (X) and directions (O) using the EM algorithm [12]. In
each cluster, the minutiae position X is modeled by a bivariate Gaussian distribution and the minutiae direction O is modeled using a Von-Mises distribution
[6]. Each minutia m(X, O) in a fingerprint of class G has the following mixture
density:
f (x, o|ΘG ) =
NG
X
τg · fX (x|µg , Σg ) · fO (o|νg , κg ),
(1)
g=1
where NG is the number of clusters in the mixture for class G, τg is the weight
for the g-th cluster, ΘG is the set of parameters describing the distributions in
each cluster, fX (x|µg , Σg ) is the p.d.f. over minutiae position in the g-th cluster,
and fO (o|νg , κg ) is the p.d.f. over minutiae direction in the g-th cluster. Minutiae in the same cluster have the same distribution of directions, establishing a
dependence between minutiae position and direction.
Let T be a fingerprint belonging to class G with minutiae density f (x, o|ΘG ).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f (x, o|ΘH ). Let m(X T , OT ) and m(X Q , OQ ) be two minutiae from T and Q,
respectively. The probability that these two minutiae would match is defined as
P (|X T − X Q | ≤ x0 , |OT − OQ | ≤ o0 |ΘG , ΘH )
=
N
G N
H
P
P
(2)
τg · τh · P (|X T − X Q | ≤ x0 |µg , µh , Σg , Σh ) · P (|OT − OQ | ≤ o0 |νg , νh , κg , κh ),
g=1 h=1
where parameters x0 = 15 pixels and o0 = 22.5 degrees are used as tolerances.
Note that this probability can be directly computed since (X T − X Q ) follows a
2D Gaussian distribution with mean (µg − µh ) and covariance (Σg + Σh ); and
(OT −OQ ) can be approximated by a Von-Mises distribution with mean (νg −νh )
and variance κg,h defined as [13]:
A(κg,h ) = A(κg )A(κh ),
A(x) = 1 − 21 − 8x1 2 − 8x1 3 + o(x−3 ).
(3)
(4)
Finally, the PRC, or the probability of matching k pairs of minutiae between
Q and T , is calculated as [6]
p(m, n, k) =
e−λ · λk
, λ = mnl,
k!
(5)
where m and n are the number of minutiae in Q and T , respectively, and
l = P (|X T − X Q | ≤ x0 , |OT − OQ | ≤ o0 |ΘG , ΘH ) is the probability calculated in Eq. (2). This PRC calculation corresponds to the Poisson probability
mass function with mean λ = mnl, which can be interpreted as the expected
number of matches from the total number of mn possible pairings between Q
and T with the probability of each match being l.
To appear at the International Conf. on Biometrics (ICB), June 2009
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Yi Chen and Anil. K. Jain
2.2
Pattern-specific Ridge Modeling
To incorporate ridge features in the model, we extend the density function in
Eq. (1) to the following mixture density for class G:
f (x, o, r, c|ΘG ) =
NG
X
τg · fX (x|µg , Σg ) · fO (o|νg , κg ) · fR (r|ωg , σg2 ) · fC (c|λg ), (6)
g=1
where fR (r|ωg , σg2 ) and fC (c|λg ) are the probability density functions of ridge
period and ridge curvature associated with each minutia in the g-th cluster, respectively. We use a Gaussian distribution to model ridge period because ridge
period only fluctuates towards the tip (smaller) and flexion crease (larger) regions. Ridge curvature, on the other hand, is usually low except near the core
and delta regions, and is modeled by a Poisson distribution. Note that minutiae
in the same cluster have the same distribution of direction, ridge period and
curvature, hence, establishing a dependency among these features.
1
2
3
2
1
3
Fig. 2. Converting local ridge structure to a canonical form such that matching two
minutiae and their associated ridges is equivalent to matching the position and direction
of minutiae and the period and curvature of the ridges.
Let T be a fingerprint belonging to class G with minutiae density f (x, o, r, c|ΘG ).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f (x, o, r, c|ΘH ). Let m(X T , OT , RT , C T ) and m(X Q , OQ , RQ , C Q ) be two minutiae from T and Q, respectively. The probability that these two minutiae would
match is then defined as
P (|X T − X Q | ≤ x0 , |OT − OQ | ≤ o0 , |RT − RQ | ≤ r0 , |C T − C Q | ≤ c0 |ΘG , ΘH ) =
N
G N
H
P
P
τg · τh · P (|X T − X Q | ≤ x0 |µg , µh , Σg , Σh ) · P (|OT − OQ | ≤ o0 |νg , νh , κg , κh )
g=1 h=1
·P (|RT − RQ | ≤ r0 |ωg , ωh , σg2 , σh2 ) · P (|C T − C Q | ≤ c0 |λg , λh ),
(7)
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length
5
where r0 = 2 and c0 = 2. Here, matching the ridges associated with two minutiae
is equivalent to matching the ridge period and curvature after converting the
ridge structure associated with each minutia to a canonical form centered at the
minutia and rotated by the minutia direction (see Fig. 2). Similar to Eq. (2),
this probability can be directly computed since (RT − RQ ) follows a Gaussian
distribution with mean (ωg − ωh ) and variance (σg2 + σh2 ) and (C T − C Q ) follows
a Skellam distribution [14] with mean (λg − λh ) and variance (λg + λh ). The
final PRC is again computed as in Eq. (5), except that l is replaced by Eq. (7).
2.3
Pore Modeling
Since pores are almost evenly spaced along the ridges [10], modeling pores can be
approximated by modeling the intra-ridge spacing of pores on modeled ridges.
As a result, each minutia and its local features (position X, direction O, ridge
period R, ridge curvature C and pore spacing S) have the following mixture
density for fingerprints from class G:
f (x, o, r, c, s|ΘG ) =
N
G
P
g=1
(8)
τg · fX (x|µg , Σg ) · fO (o|νg , κg ) · fR (r|ωg , σg2 ) · fC (c|λg ) · fS (s|µp , σp2 ),
where fS (s|µp , σp2 ) is the probability density function for pore spacing. Because
there is no evidence that intra-ridge pore spacing is dependent on minutiae
location or ridge flow patterns, pore spacing is not clustered.
Let T be a fingerprint belonging to class G with minutiae density f (x, o, r, c, s|ΘG ).
Similarly, let Q be a fingerprint belonging to class H with minutiae density
f (x, o, r, c, s|ΘH ). Let m(X T , OT , RT , C T , S T ) and m(X Q , OQ , RQ , C Q , S Q ) be
two minutiae from T and Q, respectively. The probability of matching the two
minutiae features can be calculated as follows:
P (|X T − X Q | ≤ x0 , |OT − OQ | ≤ o0 , |RT − RQ | ≤ r0 , |C T − C Q | ≤ c0 , |S T − S Q | ≤ s0
|ΘG , ΘH ) =
N
G N
H
P
P
τg · τh · P (|X T − X Q | ≤ x0 |µg , µh , Σg , Σh )
g=1 h=1
·P (|OT − OQ | ≤ o0 |νg , νh , κg , κh ) · P (|RT − RQ | ≤ r0 |ωg , ωh , σg2 , σh2 )
·P (|C T − C Q | ≤ c0 |λg , λh ) · P (|S T − S Q | ≤ s0 |σp2 ),
(9)
where s0 = 2. Again, this probability can be directly calculated since (S T − S Q )
follows a Gaussian distribution with mean 0 and variance 2σp2 . Again, PRC is
still computed as in Eq. (5), except that l is now replaced by Eq. (9). Note that
after incorporating ridge and pore features in our individuality model, PRC
represents the probability of matching k minutiae not only in their positions
and directions, but also with respect to the local ridge period and curvature as
well as pore spacing in the neighborhood of the matching minutiae.
3
Model Evaluation and Validation
In order to demonstrate the utility of our individuality model, we perform the
following evaluation protocol: (i) learn the mixture density of minutiae and their
To appear at the International Conf. on Biometrics (ICB), June 2009
6
Yi Chen and Anil. K. Jain
local ridge and pore features, (ii) compute the theoretical probability of random
correspondence (PRC), and (iii) compare the theoretical PRC with the empirical values obtained on a public database. The database used for evaluation is
the NIST Special Database 4 (NIST-4) [16], which contains 2,000 pairs (4,000
images) of inked rolled prints at 500 ppi. Rolls are appropriate for fingerprint individuality study because they provide a complete representation of fingerprints.
Each fingerprint in NIST-4 comes with a class label (28.2% right loop, 26.6%
left loop, 21.5% whorl, 19% arch and 4.7% tented arch) assigned by forensic
experts. These prints are manually aligned by the author based on the locations
of core(s). For example, fingerprints of the left loop, right loop and tented arch
classes are aligned at the core point; fingerprints of the whorl class are aligned
at the centroid of the two cores; and fingerprints of the arch class are aligned
at the highest curvature point on the most upthrusting ridge. Figure 3 shows
the empirical distribution of minutiae positions from fingerprints in each of the
five classes. It can be observed that the minutiae distribution is highly correlated
with the fingerprint class. Note the higher minutiae density in the core and delta
regions, which is consistent with Champod’s finding [8].
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
400
400
400
400
400
500
500
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500
600
600
600
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100
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(a)
400
500
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100
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(b)
400
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100
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300
(c)
400
500
600
100
200
300
(d)
400
500
600
600
100
200
300
400
500
(e)
Fig. 3. Distribution of minutiae positions (empirically extracted [20]) in each of the five
fingerprint classes (a) arch, (b) tented arch, (c) left loop, (d) right loop, and (e) whorl,
obtained from 4,000 rolled fingerprints in NIST-4. The darker the area, the higher the
minutiae density. These plots (with size 600 × 600) were smoothed using a disk filter
with a radius of 25 pixels. The center of each plot corresponds to the alignment origin.
The theoretic probability of matching a minutiae pair between impostor fingerprints of each class is calculated using Equation 2. As shown in Table 1,
impostor fingerprints from the same class have a higher matching probability
(highlighted in gray) than those from different classes. This is consistent with
the results of Jain et al.’s study [18], which revealed that fingerprints from the
same class are more likely to be matched (have higher False Accept Rate) than
fingerprints from difference classes.
For comparison, we also calculate the empirical probabilities of matching
a minutiae pair between impostor fingerprints based on NIST-4. An in-house
fingerprint matcher [15] was used to automatically establish minutiae correspondences between the impostor pairs in NIST-4. To be compatible with our model,
the matcher is restricted to register fingerprints within a 50 × 50 neighborhood
of the manually aligned origin. A total of 7, 998, 000 impostor matches were
conducted and the empirical probability of matching a minutiae pair between
600
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length
7
Table 1. Theoretical probabilities of matching a minutiae pair between impostor fingerprints belonging to class A=arch, TA=tented arch, L=left loop, R=right loop and
W=whorl based on the proposed model.
Type
A
TA
L
R
W
A
13.20 × 10−4
6.05 × 10−4
6.33 × 10−4
3.95 × 10−4
4.65 × 10−4
TA
6.05 × 10−4
12.76 × 10−4
6.85 × 10−4
5.01 × 10−4
6.40 × 10−4
L
6.33 × 10−4
6.85 × 10−4
10.95 × 10−4
7.44 × 10−4
4.92 × 10−4
R
3.95 × 10−4
5.01 × 10−4
7.44 × 10−4
11.59 × 10−4
4.74 × 10−4
W
4.65 × 10−4
6.40 × 10−4
4.92 × 10−4
4.74 × 10−4
10.01 × 10−4
Table 2. Empirical probabilities of matching a minutiae pair between impostor fingerprints belonging to class A=arch, TA=tented arch, L=left loop, R=right loop and
W=whorl based on NIST-4.
Type
A
TA
L
R
W
A
14.33 × 10−4
12.25 × 10−4
7.73 × 10−4
8.31 × 10−4
2.35 × 10−4
TA
12.25 × 10−4
12.54 × 10−4
8.33 × 10−4
9.17 × 10−4
2.61 × 10−4
L
7.73 × 10−4
8.33 × 10−4
11.46 × 10−4
5.92 × 10−4
4.36 × 10−4
R
8.31 × 10−4
9.17 × 10−4
5.92 × 10−4
11.54 × 10−4
4.64 × 10−4
W
2.35 × 10−4
2.61 × 10−4
4.36 × 10−4
4.64 × 10−4
6.36 × 10−4
two impostor prints is computed by the average of k/(m × n), where k is the
number of matched minutiae and m and n are the number of minutiae in the
two fingerprints. This probability is tabulated by the fingerprint class information, resulting in all possible intra-class and inter-class probability values (see
Table 2). Note that the theoretical probabilities based on our model are closer
to empirical probabilities (with correlation coefficient 0.59) compared to those
calculated by Zhu et al.’s model (empirical probability = 1.8 × 10−3 , theoretical
probability = 6.50 × 10−4 ) [6].
h
r
(a)
(b)
Fig. 4. Ridge features in a neighborhood (30 × 30 pixels) of a minutia. (a) ridge period
(average of inter-ridge distance h), and (b) ridge curvature (inverse of radius r).
To incorporate ridge features, we extract ridge period and curvature in a 30
× 30 neighborhood of each minutia (see Fig. 4). Ridge period is calculated as
the average inter-ridge distances, and ridge curvature, defined as the inverse of
its radius, is calculated by the second derivative of sampled ridge points. In the
model, these features are used to retrain the clustering algorithm and reevaluate
the theoretical probability in Equation 7. In the empirical case, minutiae cor-
To appear at the International Conf. on Biometrics (ICB), June 2009
8
Yi Chen and Anil. K. Jain
respondences that disagree in ridge period or curvature (with difference larger
than r0 and c0 , respectively) are removed. The resulting number of matched
minutiae pairs k 0 is used to recalculate the empirical probability as the average of k 0 /(m × n). The theoretical and empirical matching probability matrices
tabulated by fingerprint class information after incorporating the ridge features
are shown in Tables 3 and 4, respectively. As we can see, higher probabilities
are still observed among impostor fingerprints from the same class than those
from different classes. The use of ridge features reduces the probability of random minutiae correspondence both empirically and theoretically; the correlation
coefficient between the two probability matrices is increased to 0.67.
Table 3. Theoretical probabilities of matching a minutiae pair and their local ridge
features between impostor fingerprints belonging to class A=arch, TA=tented arch,
L=left loop, R=right loop and W=whorl based on the proposed model.
Type
A
TA
L
R
W
A
6.21 × 10−4
2.59 × 10−4
2.62 × 10−4
1.74 × 10−4
1.88 × 10−4
TA
2.59 × 10−4
6.06 × 10−4
3.36 × 10−4
2.40 × 10−4
2.92 × 10−4
L
2.62 × 10−4
3.36 × 10−4
5.52 × 10−4
3.71 × 10−4
2.24 × 10−4
R
1.74 × 10−4
2.40 × 10−4
3.71 × 10−4
5.99 × 10−4
2.21 × 10−4
W
1.88 × 10−4
2.92 × 10−4
2.24 × 10−4
2.21 × 10−4
4.56 × 10−4
Table 4. Empirical probabilities of matching a minutiae pair and their local ridge
features between impostor fingerprints belonging to class A=arch, TA=tented arch,
L=left loop, R=right loop and W=whorl based on NIST-4.
Type
A
TA
L
R
W
A
5.86 × 10−4
4.70 × 10−4
2.91 × 10−4
3.10 × 10−4
0.84 × 10−4
TA
4.70 × 10−4
5.68 × 10−4
3.68 × 10−4
3.89 × 10−4
1.07 × 10−4
L
2.91 × 10−4
3.68 × 10−4
5.05 × 10−4
2.39 × 10−4
1.79 × 10−4
R
3.10 × 10−4
3.89 × 10−4
2.39 × 10−4
5.12 × 10−4
1.93 × 10−4
W
0.84 × 10−4
1.07 × 10−4
1.79 × 10−4
1.93 × 10−4
2.78 × 10−4
When pores are also incorporated in the model, we reevaluate the theoretical
probability of matching a minutiae pair using Equation 9. Because pores are more
reliably captured at 1000 ppi than 500 ppi, we use the NIST-30 database [17] (720
rolled images at 1000 ppi) for extracting [19] and modeling the distribution of
pore spacing. The empirical distribution and a Gaussian fit to the distribution
with mean 12.67 and standard deviation 0.82 are shown in Figure 5. This is
consistent with the estimates of Roddy and Stosz [10], who suggested that the
most frequently observed pore spacing is 13 pixels at 1100 ppi. The theoretically
probability matrix of matching a minutiae and their local ridge and pore features
tabulated by fingerprint class information is shown in Table 5.
Having obtained the probability of matching one minutiae pair, we can calculate the PRC, or the probability of matching k minutiae pairs, given m and
n minutiae in the two fingerprints using Equation 5. For example, when only
minutiae position and orientation are incorporated, the theoretic PRC of hav-
To appear at the International Conf. on Biometrics (ICB), June 2009
Title Suppressed Due to Excessive Length
0.6
9
Empirical measure
Gaussion fit (µ = 12.67, σ = 0.82)
Density
0.5
0.4
0.3
0.2
0.1
0
6
7
8
9
10
11
12
13
14
15
16
Pore spacing
Fig. 5. Distribution of intra-ridge pore spacings (distance between two neighboring
pores on the same ridge) obtained from 720 fingerprints in NIST-30 (1000 ppi) using
an automatic pore extraction algorithm [19].
Table 5. Theoretical probabilities of matching a minutiae pair, their local ridge and
pore features between impostor fingerprints belonging to class A=arch, TA=tented
arch, L=left loop, R=right loop and W=whorl based on the proposed model.
Type
A
TA
L
R
W
A
5.68 × 10−4
2.37 × 10−4
2.40 × 10−4
1.59 × 10−4
1.72 × 10−4
TA
2.37 × 10−4
5.55 × 10−4
3.08 × 10−4
2.20 × 10−4
2.67 × 10−4
L
2.40 × 10−4
3.08 × 10−4
5.05 × 10−4
3.40 × 10−4
2.05 × 10−4
R
1.59 × 10−4
2.20 × 10−4
3.40 × 10−4
5.48 × 10−4
2.02 × 10−4
W
1.72 × 10−4
2.67 × 10−4
2.05 × 10−4
2.02 × 10−4
4.17 × 10−4
Table 6. An example comparison among PRCs obtained from Pankanti’s uniform
model, Zhu’s mixture model and the proposed model with empirical values on NIST-4.
Empirical values were calculated using Eq. (5) by substituting λ with the mean number
of matches (weighted by the class distribution) found empirically.
(m,n,k) Uniform [5] Mixture [6] Proposed Empirical
(52,52,12) 4.3 × 10−8
4.4 × 10−4 5.77 × 10−7 7.72 × 10−7
(62,62,12) 2.9 × 10−7
4.1 × 10−4 1.79 × 10−5 2.34 × 10−5
ing, say, k = 12 minutiae matches given m = 52 and n = 52 is 5.77 × 10−7 .
This is more consistent with the empirical probability (7.72 × 10−7 ) than the
previously published models (see Table 6). When ridge period and curvature are
incorporated in addition to minutiae, both theoretical and empirical probabilities (for k = 12, m = 52, n = 52) are reduced to 1.86 × 10−10 and to 5.36 × 10−11 ,
respectively. When pore spacing is incorporated, the theoretic probability drops
further to 6.93 × 10−11 .
4
Summary
We have proposed a mixture model to evaluate fingerprint individuality based on
five major fingerprint classes using minutiae and non-minutiae features. Experimental results show that the estimated matching probabilities between impostor
fingerprints are highly correlated with the fingerprint class information and, as
expected, the probabilities reduce when additional features (ridges and pores)
To appear at the International Conf. on Biometrics (ICB), June 2009
10
Yi Chen and Anil. K. Jain
are incorporated in the model. Our theoretical estimates of PRCs are consistent
with those based on empirical matching. In practice, however, both theoretical
and empirical estimates can be affected by factors such as image quality and the
feature extraction and matching algorithms. In future work, we would like to
account for these factors in our individuality model.
Acknowledgments This work was supported by research grants NIJ 2007-RGCX-K183, NIJ-2007-DN-BX-0005, and ARO W911NF-06-1-0418.
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